Idea

In the Bayesian interpretation of physics, the state of a system is not a property of reality but instead indicates an observer's knowledge about the system. A pure state gives maximal information about the system (which amounts to complete information in classical mechanics but not generally in quantum mechanics), while a mixed state is more general. A mixed state can be decomposed into a probability distribution on the space of pure states, although this decomposition is unique only for classical systems. In a frequentist interpretation of probability, a mixed state can describe only a statistical ensemble of systems; the real world is in one (generally unknown) pure state (possibly with additional hidden variables in the quantum case, depending on the interpretation of quantum physics).

States in the Schrödinger picture describe the state of the world at any given time and are subject to time evolution?, while in the Heisenberg picture a single state describes the entire history of the world.

Definitions

The precise mathematical notion of state depends on what mathematical formalization of mechanics is used.

In geometric quantization

In Hilbert-space quantum mechanics

In quantum mechanics given by a Hilbert spaceHH, a pure state is a ray in HH, which we often call the Hilbert space of states. Strictly speaking, the space of states is not HH but (H∖{0})/ℂ(H \setminus \{0\})/\mathbb{C}, or equivalently S(H)/U(1)S(H)/\mathrm{U}(1). A mixed state is then a density matrix on HH.

Arguably, the correct notion of state to use is that of quasi-state; every state gives rise to a unique quasi-state, but not conversely. However, when either classical mechanics or Hilbert-space quantum mechanics is formulated in AQFT, every quasi-state is a state (at least if the Hilbert space is not of very low dimension, by Gleason's theorem). See also the Idea-section at Bohr topos for a discussion of this point.

In FQFT

In this formulation the (n-1)-morphism in 𝒞\mathcal{C} assigned to an (n−1)(n-1)-dimensional manifoldΣn−1\Sigma_{n-1} is the space of states over that manifold. A state is accordingly a generalized element of this object.

Pure and mixed states

In statistical physics, a pure state is a state of maximal information, while a mixed state is a state with less than maximal information. In the classical case, we may say that a pure state is a state of complete information, but this does not work in the quantum case; from the perspective of the information-theoretic or Bayesian interpretation of quantum physics, this inability to have complete information, even when having maximal information, is the key feature of quantum physics that distinguishes it from classical physics.

Examples

For an impossible system, the space of states is empty; for a trivial system (with a unique way to be), then space of states is the point. This unique state is pure.

For a classical bit, a system with two distinct ways to be, the space of states is a line segment?; a state is given by a real number tt with 0≤t≤10 \leq t \leq 1. This tt is the probability that the system is in the first state, with 1−t1 - t the probability that it is in the second. The two pure states correspond to t=0t = 0 and t=1t = 1.

For a quantum bit, a qubit, the space of states is shaped like a gridiron (American or Canadian) football. A state is given by a matrix

with unit trace and nonnegative determinant; in other words, it's given by real numbers aa, bb, and cc satisfying the inequality

a2+b2+c2≤a. a^2 + b^2 + c^2 \leq a .

The pure states are those satisfying

a2+b2+c2=a, a^2 + b^2 + c^2 = a ,

forming the surface of the football (what one might call a gridiron footsphere, although properly it is a lemon). If we graph a−a2a - a^2 where it is positive (from 00 to 11) and rotate this around the aa-axis, then we get this lemon.